Capillary pinning and blunting of immiscible gravity
currents in porous media
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Citation
Zhao, Benzhong, Christopher W. MacMinn, Herbert E. Huppert,
and Ruben Juanes. “Capillary Pinning and Blunting of Immiscible
Gravity Currents in Porous Media.” Water Resour. Res. 50, no. 9
(September 2014): 7067–7081. © 2014 American Geophysical
Union
As Published
http://dx.doi.org/10.1002/2014WR015335
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American Geophysical Union (Wiley platform)
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Thu May 26 18:50:25 EDT 2016
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http://hdl.handle.net/1721.1/101639
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RESEARCH ARTICLE
10.1002/2014WR015335
Key Points:
We study experimentally the impact
of capillarity on a buoyant gravity
current
We show that capillary pressure
hysteresis can stop migration of the
current
Capillary pinning can be an effective
trapping mechanism in CO2
sequestration
Capillary pinning and blunting of immiscible gravity currents in
porous media
Benzhong Zhao1, Christopher W. MacMinn2, Herbert E. Huppert3,4,5, and Ruben Juanes1
1
Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts,
USA, 2Department of Engineering Science, University of Oxford, Oxford, UK, 3Department of Applied Mathematics and
Theoretical Physics, Institute of Theoretical Geophysics, University of Cambridge, Cambridge, UK, 4School of Mathematics,
University of New South Wales, Kensington, New South Wales, Australia, 5Faculty of Science, University of Bristol,
Bristol, UK
Abstract Gravity-driven flows in the subsurface have attracted recent interest in the context of geologiSupporting Information:
Readme
Video of micromodel simulation
Video of micromodel experiment
Correspondence to:
R. Juanes,
juanes@mit.edu
Citation:
Zhao, B., C. W. MacMinn, H. E. Huppert,
and R. Juanes (2014), Capillary pinning
and blunting of immiscible gravity
currents in porous media, Water
Resour. Res., 50, 7067–7081,
doi:10.1002/2014WR015335.
Received 22 JAN 2014
Accepted 1 AUG 2014
Accepted article online 5 AUG 2014
Published online 3 SEP 2014
cal carbon dioxide (CO2) storage, where supercritical CO2 is captured from the flue gas of power plants and
injected underground into deep saline aquifers. After injection, the CO2 will spread and migrate as a buoyant gravity current relative to the denser, ambient brine. Although the CO2 and the brine are immiscible,
the impact of capillarity on CO2 spreading and migration is poorly understood. We previously studied the
early time evolution of an immiscible gravity current, showing that capillary pressure hysteresis pins a portion of the macroscopic fluid-fluid interface and that this can eventually stop the flow. Here we study the
full lifetime of such a gravity current. Using tabletop experiments in packings of glass beads, we show that
the horizontal extent of the pinned region grows with time and that this is ultimately responsible for limiting the migration of the current to a finite distance. We also find that capillarity blunts the leading edge of
the current, which contributes to further limiting the migration distance. Using experiments in etched
micromodels, we show that the thickness of the blunted nose is controlled by the distribution of porethroat sizes and the strength of capillarity relative to buoyancy. We develop a theoretical model that captures the evolution of immiscible gravity currents and predicts the maximum migration distance. By applying this model to representative aquifers, we show that capillary pinning and blunting can exert an
important control on gravity currents in the context of geological CO2 storage.
1. Introduction
Gravity currents refer to the predominantly horizontal gravity-driven flow of two fluids with different densities. Gravity currents are prevalent in nature and they occur in the atmosphere, the ocean, and the subsurface [Huppert, 2006]. Gravity currents in porous media have been well studied in the past in the context of
miscible fluids [Barenblatt, 1952; Bear, 1972; Huppert and Woods, 1995]. Results from these studies have
become important predictors for a wide range of processes such as seawater intrusions into freshwater
aquifers [Lee and Cheng, 1974; Frind, 1982] and drilling-fluid migration into surrounding reservoirs [Dussan
and Auzerais, 1993]. More recently, the study of gravity currents in porous media has provided new insights
into the storage of carbon dioxide (CO2) in deep geological reservoirs such as saline aquifers [Nordbotten
et al., 2006; Bickle et al., 2007; Hesse et al., 2007; Juanes et al., 2010; MacMinn et al., 2011]. Because CO2 is less
dense than the ambient brine in those reservoirs, the CO2 will migrate upward due to buoyancy and spread
along the top boundary of the aquifer as a gravity current.
While capillarity is absent in miscible gravity currents, it plays an important role in gravity currents with
immiscible fluids. Immiscible gravity currents are relevant in processes such as the migration of dense nonaqueous phase liquid (DNAPL) contamination in groundwater aquifers [Illangasekare et al., 1995; Pennell
et al., 1996; Ewing and Berkowitz, 2001], as well as CO2 storage in deep saline aquifers [Hesse et al., 2008;
MacMinn et al., 2010, 2011; Gasda et al., 2011]. A well-studied effect of capillarity in immiscible gravity currents is residual trapping, where tiny blobs of nonwetting fluid become surrounded by wetting fluid and
become immobilized in the pore space [Juanes et al., 2006; Hesse et al., 2008; Juanes et al., 2010; MacMinn
et al., 2010]. In addition to residual trapping, capillary forces create a partially saturated layer between the
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buoyant fluid and the ambient
fluid: the capillary fringe.
Continuum-scale models predict
that capillary effects result in a
thick nonwetting current with a
blunt nose [Lake, 1989; Nordbotten and Dahle, 2011; Golding et al.,
2013].
Zhao et al. [2013] performed the
first laboratory experiments of
immiscible gravity-exchange flow
of two immiscible fluids in
porous media. Focusing on the
early time evolution of the macroscopic interface between the
fluids, they observed a phenomenon they coined capillary pinning, where a portion of the
Figure 1. Gravity-driven flow of a buoyant, nonwetting fluid (air) over a dense, wetting fluid
macroscopic fluid-fluid interface
(propylene glycol) in a packing of glass beads. (a and b) Starting with a vertical interface
between the fluids, the flow first undergoes a lock-exchange process. (c and d) The process
remains stationary (Figures 1a
models a finite-release problem after the dense fluid reaches the left boundary. In contrast to
and 1b). They identified porethe finite release of a miscible current that spreads indefinitely, spreading of an immiscible
scale capillary pressure hysteresis
current stops at a finite distance. Dashed black box highlights the blunt nose of the current,
which has thickness hn (see section 3.2). Red lines represent predictions from our macrobetween drainage and imbibition
scopic sharp-interface model (see section 4).
as the mechanistic cause of the
pinned interface. While Zhao
et al. [2013] described the early time evolution of immiscible gravity currents (i.e., exchange flow), the
late-time behavior of immiscible gravity currents has not been studied experimentally.
Here, we study the entire evolution of immiscible gravity currents via release of a finite volume of buoyant nonwetting fluid into a dense wetting fluid in a vertically confined aquifer. In this case, the imbibition front (the portion of the interface where the wetting fluid is advancing) interacts with the left
boundary early and begins to rise (Figure 1c). The upward motion of the imbibition front causes an
increasing portion of the drainage front (the portion of the interface where the wetting fluid is retreating) to be pinned. Eventually, the entire drainage front is pinned and the nonwetting current is arrested
at a finite distance (Figure 1d). We study this system experimentally using analog fluids in flow cells
packed with glass beads, as well as in micromodels etched with small cross-stream posts. In contrast to
previous studies on immiscible gravity currents, which incorporate capillarity at the continuum scale
through a capillary pressure-saturation curve, our experiments are aimed at understanding the physical
mechanisms acting at the pore scale. We show how pore-scale mechanisms alter the behavior of immiscible gravity currents at the macroscopic scale. We extend the classical model of gravity currents in
porous media [Huppert and Woods, 1995; Hesse et al., 2007] to include capillary pinning and nose blunting and demonstrate how both mechanisms limit the spreading of a nonwetting current.
2. Laboratory Experiments in Porous Media
In this section, we present our observations from laboratory experiments in porous media. In the next
section, we connect these macroscopic observations with pore-scale mechanisms. We consider the
instantaneous release of a finite volume of buoyant nonwetting fluid into a horizontal aquifer filled
with a relatively dense wetting fluid. We conduct these experiments in a rectangular, quasi twodimensional flow cell packed with approximately monodisperse spherical glass beads, following the
same procedure described in Zhao et al. [2013]. The cell is 5.2 cm tall and 56 cm long, with a thickness
of 1 cm. We use air as the buoyant nonwetting fluid, paired with one of three dense wetting fluids
(Table 1). We report the measured porosity and effective permeability of the bead packs in Table 2. We
release the same volume of air in every experiment, fixing the width of the initial release, l 5 4.9 cm
(Figure 2).
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2.1. Capillary Pinning and
Depinning
Ambient Fluid
q ðg=cm3 Þ
l ðg=cm sÞ
c ðdyn=cmÞ
Starting from the initial condiSilicone oil
0.96
0.48
20
tion of a vertical interface sepaPropylene glycol
1.04
0.46
36
rating the buoyant, nonwetting
Glycerol-water mixture
1.2
0.47
65
fluid and the dense, wetting
fluid, the two fluids first undergo
a gravity-exchange flow. During this early period, we observe that a portion of the initial interface is pinned
and does not experience any macroscopic motion (Figures 1a and 1b). The pinning of the initial interface
during the exchange flow was described by Zhao et al. [2013]. Here we study the flow behavior after the
end of the exchange flow, at which point the wetting fluid reaches the left boundary of the flow cell and
the imbibition front starts to rise (Figures 1c and 1d). The rise of the imbibition front is accompanied by
depinning at the bottom of the vertical pinned interface. As the imbibition front rises, an increasing portion
of the drainage front is pinned. Eventually, the spreading of the gravity current stops when the entire drainage front is pinned. This is in stark contrast with miscible gravity currents, which spread indefinitely [Huppert
and Woods, 1995; Hesse et al., 2007].
Table 1. Properties of the Three Ambient Fluids Used in the Experiments
2.2. Nose Blunting
In miscible gravity currents between a buoyant, less viscous fluid and a dense, more viscous fluid, the buoyant fluid spreads out as a thin tongue, with nose thickness on the scale of one grain diameter [MacMinn and
Juanes, 2013]. In immiscible gravity currents, in contrast, we observe a blunted nose at the front of the
buoyant, nonwetting current, as well as a thick current profile (Figure 1d).
3. Micromodel Experiments and Pore-Scale Mechanisms
In this section, we connect the macroscopic observations from the previous section with their pore-scale
origins.
3.1. Capillary Pinning and Depinning
Zhao et al. [2013] studied the early stage (exchange flow) of immiscible gravity currents in porous media
and found that the strength of capillary pinning, as measured by the length of the pinned portion of the
interface, is a function of the relative importance between capillarity and gravity, as described by the
inverse of the Bond number,
Bo21 5
c=d
;
DqgH
(1)
where c is the interfacial tension between the fluids, d is the characteristic grain size of the bead pack, Dq is
the density difference between the fluids, g is the gravitational constant, and H is the height of the flow cell
(Figure 3).
To understand the mechanistic cause of the pinned interface, we conduct experiments with air and silicone
oil in micromodels made of thin acrylic plates etched with cylindrical posts on a rectangular lattice (Figure
4). The micromodels serve as a porous medium analog in the sense of introducing microstructure, while
permitting clear visualization of the flow at the pore level. In particular, the pore level fluid-fluid interface is
a direct indication of the magnitude of capillary pressures locally. The capillary pressure at each pore throat
21
is given by the Laplace pressure Pc 5c R21
, where R1 is the radius of curvature of the fluid-fluid inter1 1R2
face in the plane orthogonal to the axes of the cylindrical posts and R2 is the radius of curvature of the
fluid-fluid interface in the quasi
two-dimensional plane. The
Table 2. Properties of the Three Bead Packs Used in the Experiments
transition of the pore-scale
Bead Size (cm)
d ðcmÞ
/ð2Þ
k ðcm2 Þ
radius of curvature along the
0:0820:12
0.1
0.407
0:9 3 1025
pinned vertical interface clearly
0:120:15
0.125
0.412
1:3 3 1025
indicates the presence of capil25
0:1520:2
0.175
0.437
4:5 3 10
lary pressure hysteresis between
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drainage and imbibition (Figure
4a). Along the vertical pinned
interface, the capillary pressure
decreases with depth from the
drainage capillary pressure Pcdr to
the imbibition capillary pressure
Pcimb . This decrease in capillary
pressure, DPc 5Pcdr 2Pcimb > 0, is
Figure 2. Schematic of the laboratory experiments in porous media. We conduct experioffset by the increase in hydroments in quasi two-dimensional flow cells packed with glass beads with an initial release
static pressure along the vertical,
of a rectangular volume of buoyant, nonwetting fluid (gray) into a dense, wetting fluid
(white).
pinned interface. The balance
between these two changes in
pressure implies that the height of the pinned interface is given by Dhc 5DPc =Dqg. Since Pcdr ; Pcimb c=d; D
hc =H scales linearly with Bo21 5ðc=dÞ=ðDqgHÞ (Figure 3).
After the dense wetting current reaches the left boundary of the flow cell, the bottom of the vertical pinned
interface starts to depin and becomes part of the imbibition front. This results in a reduced vertically pinned
interface. Since the amount of capillary pressure hysteresis in the system remains the same, the hydrostatic
pressure increase along the now reduced vertical pinned interface only accounts for part of the entrypressure difference between drainage and imbibition. At the pore scale, this is reflected by the larger radius
of curvature at the top of the vertical pinned interface compared to that at the end of the exchange flow
stage (Figure 4b). To balance capillary pressure hysteresis, the pinned interface moves upward and extends
into the drainage front. The depinning of the previously pinned interface and the pinning of the previously
draining interface are synchronized with each other such that the height difference between the top and
the bottom of the pinned interface is always the same, Dhc . Eventually, the current stops when the entire
drainage front becomes pinned.
3.2. Nose Blunting
We measure the thickness of the leading edge, or ‘‘nose,’’ of the nonwetting current in the porous media
experiments, hn, and find that for a given bead pack, hn increases with Bo21. To expand on this observation, we conduct experiments in the micromodel used in Zhao et al. [2013]. The micromodel consists of
evenly spaced, uniformly sized cylindrical posts such that the pore-throat size is the same everywhere. We
will refer to this micromodel as
the ‘‘uniform micromodel.’’ We
conduct gravity current experiments under a wide range of
Bo21 by tilting the micromodel
about the x axis, thus changing
the effective strength of gravity
in the y-z plane. We do not
observe nose blunting in the uniform micromodel even when the
effective gravity is low compared
to capillarity (i.e., large Bo21;
Figure 5a).
Figure 3. Scaling of the pinned interface height and the nose thickness. In contrast to
miscible gravity currents, which spread indefinitely, immiscible gravity currents stop at a
finite distance. We measure the height difference between the highest point and the
lowest point of the nonwetting current, Dhc (inset), after the current stops and find that
it scales linearly with the strength of capillary relative to gravity, as measured by Bo21,
normalized by the height of the cell H (black circles). The slope of this linear scaling
matches that of the capillary height reported in Zhao et al. [2013] (gray circles), defined
as the normalized difference between the lower hinge height h0 and the height of the
miscible tilting point, hs (inset).
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The absence of nose blunting in
the uniform micromodel suggests that Bo21 alone is not sufficient to explain nose blunting
and that some additional parameter, possibly the pore geometry, is also important. To test
this hypothesis, we perform the
same experiments in a micromodel with anisotropic pore-throat
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sizes such that the horizontal pore throats are larger
than those in the vertical
direction. We will refer to
this micromodel as the ‘‘anisotropic micromodel.’’ We
observe nose blunting in
the anisotropic micromodel
(Figure 5b) and find that the
nose thickness increases
with increasing Bo21. The
presence of nose blunting in
the anisotropic micromodel
is similar to that of the vertical pinned interface in the
sense that along the
blunted nose, the drainage
capillary pressure transitions
from the entry pressure
Figure 4. Visualization and physical mechanism of capillary pinning. (a) Snapshot of the
needed to invade the
exchange flow phase of an immiscible gravity current between air and silicone oil in a microsmaller, vertical pore throats
model with uniform pore-throat size (a 5 1 mm). Capillary pressure hysteresis between drainat the top to the entry presage (green) and imbibition (blue) is visible via the increase in the radius of curvature of the
pore-scale fluid-fluid interface along the vertical interface. The portion of the interface
sure needed to invade the
between drainage and imbibition is pinned and does not move during the exchange flow. (b)
larger, horizontal pore
Snapshot of the finite-release phase of the same gravity current. During the finite-release
throats at the bottom. This
phase, the imbibition front rises, which leads to a reduced vertical interface. The radius of curvature of the fluid-fluid interface at the top of the vertical interface has increased, indicating
transition in capillary presthat it is now pinned (red). The reduction in the vertical pinned interface is offset by the extensure is balanced by the
sion of the pinned interface to the right, pinning increasingly larger portion of the drainage
hydrostatic pressure increase
front. The current stops when the entire drainage front is pinned.
across the blunted nose and
is visible via the change in the radius of curvature of the pore-scale fluid-fluid interface along the nose
(Figure 5b).
To draw the connection with the observations in porous media, we perform experiments in micromodels with disorder in the microstructure (Figure 6). Here we randomly assign the pore-throat radii by
drawing from a uniform distribution with a range of a 2 ½0:25; 0:75 mm. We will refer to this micromodel as the ‘‘random micromodel.’’ In this system, even though the pore-throat sizes are random and anisotropy is absent, we still observe a blunted nose that thickens with increasing Bo21 . In addition, the
disorder in the pore-throat sizes leads to a rough drainage front (Figure 6), which is also observed in
the porous media experiments.
We now derive a quasi static, invasion percolation-type model [Wilkinson and Willemsen, 1983; Lenormand et al., 1988; Cieplak and Robbins, 1990] to numerically simulate the drainage sequence in the
random micromodel. In order for a fluid-fluid interface at a given pore throat to advance to the
adjacent pore, the air pressure must be greater than the sum of the local drainage capillary entry
dr
pressure and the hydrostatic pressure in the liquid Pa > Pc;e
1Pl . Since the contact angle between silicone oil and air on an acrylic substrate is zero, the drainage capillary entry pressure is
dr
Pc;e
2cða21 1d21 Þ, where a is the width of the throat between neighboring posts and d is the
height of the posts [Lenormand et al., 1983]. We assume that the air pressure is the same everydr
where and that the oil pressure is hydrostatic. At each step, we calculate Pc;e
1Pl at every fluid-fluid
dr
interface and advance the interface with the smallest Pc;e 1Pl to the neighboring pore.
We find that our simple model is able to accurately capture the drainage sequence in the random micromodel experiments at late times, when viscous effects are small (Figure 6). The relative importance
between capillary forces and viscous forces is described by the capillary number Ca5lU=c, where l is the
viscosity of the dense, more viscous fluid, and we use the rate of nose advancement as the characteristic
velocity U. As the viscous effects become small, drainage of the nonwetting phase is completely
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Figure 5. Visualization and mechanistic origin of nose blunting. (a) Snapshot of an immiscible
gravity current between air and silicone oil in a micromodel with uniform pore-throat size
(a 5 1 mm). A significant portion of the initial interface is pinned, indicating strong capillarity
with respect to gravity. Yet the nose of the nonwetting current spreads only along the top row
of the micromodel. (b) Snapshot of an immiscible gravity current experiment between air and
silicone oil in a micromodel with anisotropic pore-throat size distribution. The pore throats
parallel to the x axis are 2 mm wide, while the pore throats parallel to the z axis are 1.5 mm
wide. The anisotropic pore-throat size distribution creates anisotropy in the capillary entry
pressure. Along the nose, the capillary pressure transitions hydrostatically from the larger capillary entry pressure required for air to drain forward, to the smaller capillary entry pressure
required for air to drain downward. This is reflected by the change in radius of curvature of
the pore-scale fluid-fluid interface along the nose.
Figure 6. Thickness and roughness of the nose of a gravity current. (a) The nose of an immiscible gravity current experiment of air spreading over silicone oil in a thin acrylic cell with round
posts etched on a rectangular grid. The pore-throat sizes are randomized and follow a uniform
distribution within the range of a 2 ½0:25; 0:75 mm. (b) We simulate the drainage process of
the micromodel experiment using the quasi-static model. The quasi-static model is able to
accurately capture the drainage process of the gravity current at late times, when viscous
effects are negligible (Ca50:0024). Here the differences in capillary pressures at the drainage
front are balanced by gravity, resulting in a characteristic thickness of the nonwetting current,
even though the interface between the fluids is rough. Videos of the micromodel experiment
and the quasi-static model simulation are provided in the supporting information.
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controlled by capillarity and
gravity. This regime is characterized by previous studies
as gravity-capillary equilibrium [Lake, 1989; Zhou et al.,
1997; Yortsos, 1995; Golding
et al., 2011; Nordbotten and
Dahle, 2011; Golding et al.,
2013]. Since the quasi-static
model only takes into
account of the local capillary
entry pressure and the
hydrostatic pressure, its ability to predict the drainage
sequence shows that nose
blunting is strongly influenced by the spatial distribution of pore-throat sizes and
hence, different capillary
entry pressures: forward
drainage of the nonwetting
fluid is temporarily stalled at
narrow pore throats, as
downward drainage through
wider pore throats becomes
more favorable. Videos
showing the comparison
between a random micromodel experiment and the
quasi-static model simulation are provided in the supporting information. We find
that the anisotropy of the
pore space has negligible
effects on the imbibition
front. This is because the
imbibition front advances
through cooperative invasion of multiple pores simultaneously, whereas the
drainage front relies on subsequent invasion of individual pores [Cieplak and
Robbins, 1990].
Since nose blunting is
caused by the spatial distribution of capillary entry
pressures, we can define
a characteristic maximum
max
capillary entry pressure Pc;e
and a characteristic minimum capillary entry
min
pressure Pc;e
. The difference
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between the two characteristic capillary entry pressures is balanced by the increase in hydrostatic pressure along the nose of the current and the nose thickness is given by
hn 5
max
min
Pc;e
2Pc;e
1ho ;
Dqg
(2)
where ho is some base nose thickness, which is on the order of a single bead. We scale hn and ho by the cell
0
0
max
min
height such that h n 5hn =H; h o 5ho =H, and Pc;e
and Pc;e
by the Laplace pressure such that
0 dr
0 dr
max
21
min
21
P c;max 5Pc;e =ðcd Þ and P c;min 5Pc;e =ðcd Þ. We obtain
0
h n5
c=d
ho
0 dr
0 dr
2P c;min Þ1 :
ðP
H
DqgH c;max
(3)
0 dr
0 dr
The first term on the right-hand side of equation (3) is equal to Bo21 f, where we define f5P c;max 2P c;min ,
which measures the amount of disorder in the pore-throat size distribution. A larger value of f corresponds
to a wider pore-throat size distribution, and hence, a more blunted nose for a given Bo21. For aquifers
0
where H d, ho becomes negligible and the dimensionless nose height h n is given by
0
h n Bo21 f:
(4)
4. Macroscopic Sharp-Interface Model
We propose a macroscopic model that predicts the evolution, and eventually the stopping of immiscible
finite-release gravity currents. We base our model on the classical sharp-interface model [Huppert and
Woods, 1995; Yortsos, 1995; Hesse et al., 2007], which assumes hydrostatic pressure distribution everywhere.
The classical formulation describing miscible gravity currents [Huppert and Woods, 1995] is a partial differential equation for the height of the fluid-fluid interface, and is given by
@h
@
@h
2j
ð12f Þh
50;
(5)
@t
@x
@x
where h(x, t) is the height of the fluid-fluid interface measured from the bottom of the aquifer,
j5Dqgk=ðl/Þ is the characteristic buoyancy velocity, and f(h) is the so-called fractional flow function
and is given by
f5
h
;
h1MðH2hÞ
(6)
with M being the viscosity ratio between the fluids.
The classical formulation can be extended to include capillary effects based on the concept of macroscopic capillarity [Nordbotten and Dahle, 2011; Golding et al., 2011, 2013], where one assumes that capillary pressure is a function of the nonwetting phase saturation [Leverett, 1941]. To relate fluid saturation
to capillary pressure, they use empirical models for capillary pressure curves such as the Brooks-Corey
model [Brooks and Corey, 1966] and the van Genuchten model [van Genuchten, 1980]. They further
assume gravity-capillary equilibrium (@Pc =@z5Dqg) to relate fluid saturation and height of the fluid-fluid
interface. The gravity-capillary equilibrium assumption predicts a capillary fringe, or transition zone,
where the nonwetting phase saturation varies considerably with respect to depth. The resulting formulation is a vertically integrated model for the interface height that reduces to the classical sharp-interface
model for single-phase gravity current in the absence of capillarity. This type of model shows that the
existence of a capillary fringe could significantly impact the migration of two-phase gravity currents, and
it predicts that capillary effects create a thick current as well as a rounder nose profile [Nordbotten and
Dahle, 2011; Golding et al., 2011, 2013]. These models, however, assume that capillary pressure is a
unique function of the nonwetting phase saturation only, and therefore, do not capture the hysteretic
nature of capillary pressure. As a result, they do not reproduce interface pinning and finite spreading of
two-phase gravity currents.
Capillary pressure hysteresis has been recently incorporated in vertically integrated models of two-phase
flow in porous media [Zhao et al., 2013; Doster et al., 2013]. Doster et al. [2013] study the impact of
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capillary pressure hysteresis and
residual trapping on the constitutive parameter functions of the
vertically integrated models.
Zhao et al. [2013] study twophase exchange flow in horizontal porous layers. They incorporate capillary pressure hysteresis
Figure 7. Macroscopic sharp-interface model for immiscible gravity currents that
to the classical sharp-interface
includes capillary pinning and blunting. After the dense, wetting fluid hits the left
model by introducing a capillary
boundary, the imbibition front (blue line) starts to move up, causing the active drainage
term in the flux function that
front (green line) to become progressively pinned. As a result, the length of the pinned
interface (red line) grows with time, such that the height difference between the imbibicaptures capillary pinning. The
tion front and drainage front is always Dhc . The buoyant current stops spreading when
resulting model reproduces the
the entire drainage front becomes pinned. The model also enforces the leading edge of
pinned interface as well as the
the buoyant current to fill downward to the characteristic nose height hn before advancing forward. We solve the model numerically using a finite volume scheme. This figure
early time spreading dynamics.
illustrates how we incorporate the pinned interface into our numerical scheme as
Here, we extend this model to
explained in Appendix A, with solid circles being cell centers and dashed line being cell
an inclined aquifer and we
interfaces.
include blunting at the nose of
the current as well as progressive pinning of the drainage front, which is ultimately responsible for stopping the migration of the gravity current.
4.1. Mathematical Model
We consider an immiscible gravity current between a buoyant nonwetting fluid with density q and a dense
wetting fluid with density q1Dq in an inclined porous medium of thickness H and slope h (Figure 7). We
assume that the porous medium is homogeneous and isotropic with permeability k and porosity / and that
the boundaries of the flow domain are impermeable. Since we assume that the two fluids are separated by
a sharp interface, the thickness of the fluid layers must sum to the thickness of the porous layer everywhere,
h1 1h2 5H. By assuming hydrostatic pressure in both fluids, we can express the pressure distribution in the
layer as
(
for z > h2
PI 2qgcos hðz2h2 Þ
P5
;
(7)
PI 2Pc 1ðq1DqÞgcos hðh2 2zÞ for z h2
where PI is the unknown pressure at the interface and g is the gravitational acceleration. By definition, the
pressure difference across the interface between the nonwetting fluid and the wetting fluid is the capillary
pressure Pc.
The volumetric flux per unit width of fluid phase i along the slope is given by Darcy’s law
qi 52kki ð@P=@x2qi gsin hÞ, where ki 5kri =li and qi are the mobility and density of the fluid phase, and
kri is the relative permeability to that phase. Since we assume the two fluid phases to be completely segregated and the fluid saturation to be homogeneous within each layer, the relative permeabilities are
constant. The flow rate is given by the product of the thickness of the fluid phase and its volumetric
flux, Qi 5hi qi ,
@PI
@h1
2qg cos h
2sin h ;
@x
@x
@PI @Pc
@h2
Q2 52h2 kk2
2
2ðq1DqÞg cos h
2sin h :
@x
@x
@x
Q1 52h1 kk1
(8a)
(8b)
We solve for @PI =@x by imposing global volume conservation Q1 1Q2 50. We then substitute @PI =@x into
equation (8) and, enforcing the identity h1 1h2 H, we can express the flow rates in terms of only h2, which
also represents the height of the interface:
Q2 5
ZHAO ET AL.
Dqgk
@h2
@Pc =Dqg
ð12f Þh2 cos h
1sin h2
;
@x
l2
@x
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f5
h2
;
h2 1MðH2h2 Þ
(9b)
where f is the fractional flow function and M5k1 =k2 is the mobility ratio. To obtain an equation for the evolution of the interface, we consider the conservation of volume of the dense fluid over region Dx and time
Dt. The change in volume of the dense fluid is given by
DV2 5Dh2 Dxð12Swc Þ/5ðQ2 jx1Dx 2Q2 jx ÞDt;
(10)
where Swc is the connate water saturation, which accounts for the fact that the nonwetting current does not
displace all the wetting fluid on the drainage front. Inserting equation (9) into equation (10) and taking limits for small Dx and Dt, we obtain the partial differential equation for the evolution of the interface height
h h2 :
@h
@ ð12f Þ
@h
@hc
h
50;
(11)
2j
1tan h2
@x
@t
@x ð12Swc Þ @x
where we define j5Dqgcos hk=ðl2 /Þ as the characteristic buoyancy velocity and hc 5Pc =ðDqgcos hÞ as the
characteristic height of capillary pressure hysteresis.
It remains to write the capillary pressure, Pc, in terms of the evolution of the interface. Along the imbibition
front, we take the capillary pressure to be constant and equal to a characteristic imbibition capillary pressure, Pcimb . Along the drainage front (except for the nose region, which we handle separately), we take the
capillary pressure to be constant and equal to a characteristic drainage capillary pressure, Pcdr . Along the
pinned interface, which connects the imbibition front to the drainage front, the capillary pressure transitions accordingly from its imbibition value to its drainage value. The transition in capillary pressure is offset
by the change in hydrostatic pressure along the pinned interface, with a corresponding difference in interface height of Dhc 5ðPcdr 2Pcimb Þ=Dqgcos h between the two ends of the pinned interface. We assume a
similar transition at the nose, which we define as the region where the current is thinner than the nose
min
max
height (H2h < hn ), and within which the capillary pressure transitions from Pc;e
to Pc;e
with a corresponding change in thickness hn 2h0 (see section 3.2). We then have that @hc =@x50 along the imbibition drainage front and along the drainage front, whereas @hc =@x5@h=@x along the pinned interface and at the
nose.
We write the model in dimensionless form by scaling h, Dhc , and x by the characteristic length H and t by
the characteristic time T5H=j, to obtain a partial differential equation describing the dimensionless interface height with respect to dimensionless time
@h @ ð12f Þ
@h
@hc
h
50;
2
1tan h2
@x
@t @x ð12Swc Þ @x
(12)
where we substitute each variable in equation (11) with its dimensionless counterpart.
5. Spreading Dynamics
Hesse et al. [2007] developed early and late-time similarity solutions for the spreading of a miscible gravity
current in a horizontal porous layer. They found that the tip of the buoyant current (the nose position) prop1
agates as xn t 2 during early times, when the released fluid fills the entire height of the aquifer, and later
1
as xn t3 , when the height of the released fluid is much smaller than the thickness of the aquifer. They
found that the time it takes for the late-time scaling of the nose position to become valid (the transition
time) is a function of the mobility ratio M, such that the transition time is longer with higher M.
1
The nose positions measured from our experiments initially follow the same xn t2 scaling as the early
1
time behavior of their miscible counterpart [Hesse et al., 2007] (Figure 8). The xn t 3 scaling is absent
because of the large mobility ratios in our experiments (M 2500) and the relatively large value of Bo21 ,
1
such that current migration is stopped before the transition from early time scaling (xn t 2 ) to late-time
1
1
scaling (xn t 3 ) is reached. Using our macroscopic sharp-interface model, we find that the xn t3 spreading
21
regime does exist in immiscible gravity currents, but only for those at small Bo and small M, after a
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sufficiently long time. For
immiscible gravity currents at
high Bo21, the initial vertical
interface never gets completely depinned and, hence,
the buoyant current never gets
truly thin compared to the
height of the aquifer—as a
result, it never enters the xn
1
t 3 spreading regime.
Figure 8. Spreading dynamics of immiscible gravity currents in horizontal porous layers.
Time evolution of the nose positions of air spreading over silicone oil (blue circles;
Bo21 50:033), propylene glycol (green circles; Bo21 50:054), a glycerol-water mixture (red
1
circles; Bo21 50:085) show immiscible gravity currents initially follow the same xn t 2 scaling as their miscible counterparts. The stopping distance of the buoyant current as well as
its rate of advancement decreases with increasing Bo21. The gray dashed line represents
the right boundary of the flow cell. The solid lines represent the simulated nose positions
from the sharp-interface model. The apparent long-time scale adjustments of the nose positions after complete pinning of the gravity current are related to throat widening due to the
slow gravity drainage of the wetting films coating the glass beads in the drained region,
which effectively decreases the value of Dhc .
Although capillarity does not
change the scaling of the nose
positions with respect to time,
it does cause a reduction in the
rate of propagation of the
gravity current: the speed of
propagation decreases with
higher Bo21 (Figure 8).
The most striking feature in the
spreading dynamics of an
immiscible gravity current is
that the current stops at a
finite distance, as opposed to a
miscible current, which in the sharp-interface limit continues to spread forever. The mechanistic cause of
the finite migration distance is capillary pressure hysteresis: the current stops when the height difference
between the imbibition front and the drainage front reaches Dhc 5DPc =Dqg. As a result, the stopping distance xf is a monotonically
decreasing function of Bo21.
The relationship between xf
and Bo21, however, is markedly nonlinear. At sufficiently
low Bo21, the entire initial vertical interface depins and the
imbibition front expands laterally outward. This allows the
wetting fluid to displace a
much larger volume of the
nonwetting fluid that would
otherwise be ‘‘trapped’’ by the
vertical pinned interface at
higher Bo21. As a result, xf
increases significantly at low
Bo21 such that it approaches
infinity when capillarity is negFigure 9. Stopping distance of an immiscible gravity current. The stopping distance is conligible (Figure 9). In addition to
trolled by both capillary pinning and nose blunting. While both processes depend on Bo21,
Bo21, xf is also a function of
nose blunting is also strongly influenced by the pore geometry. Here we plot the dimen21
the pore-throat size distribusionless stopping distance xf =H as a function of Bo of a square-shaped release of nonwetting fluid. The stopping distances are obtained from our macroscopic sharp-interface
tion of the porous medium, as
model, for M52500 and three different f values. As expected, the stopping distance
characterized by the ‘‘poreapproaches infinity as capillarity becomes negligible and it tends to zero as capillarity is
scale disorder’’ parameter f
increased to the limit of complete pinning. The ‘‘kink’’ in the curves above occurs at sufficiently small Bo21 such that the initial vertical interface completely depins and the imbibi(section 3.2). Higher f repretion front is able to expand outward before the current stops. The final state of the current
sents a wider pore-throat size
21
at large and small Bo is illustrated in the figure inset. The vertical dashed line in the inset
distribution, which causes
shows the initial width of the nonwetting fluid.
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Table 3. Geologic Formations Associated With Actual CO2 Sequestration Projects, to Which We Apply Our Model
Aquifer
Otway
k ðm2 Þ
H ðmÞ
25
Tuscaloosa
60
Mt. Simon
Teapot Dome
100
30
7310
213
2.2310
213
1310213
3310214
Sgr (–)
Swc (–)
Bo21 (–)
C (–)
References
23
0.33
Hortle et al. [2009] and
Underschultz et al. [2011]
Kuuskraa et al. [2009] and
Krevor et al. [2012]
Szulczewski et al. [2012]
Garcia [2005] and
Chiaramonte et al. [2008]
0.33
0.09
9:9310
0.31
0.05
7:431023
0.37
0.3
0.3
0.4
0.4
6:631023
4:031022
0.5
0.5
more vertical drainage as the nonwetting fluid spreads outward. The increase in vertical extent of the nonwetting current reduces its lateral extent, leading to a shorter stopping distance (Figure 9).
6. Application to Carbon Sequestration
Geologic CO2 sequestration relies on trapping mechanisms to securely store the injected CO2. These mechanisms include residual trapping, where tiny blobs of CO2 are immobilized by capillary forces [IPCC, 2005;
Juanes et al., 2006], and solubility trapping, where CO2 dissolves into the ambient groundwater [IPCC, 2005].
Recent studies have shown that convective dissolution, where free-phase CO2 is carried away from the
buoyant CO2 current, can greatly enhance the rate of solubility trapping [Weir et al., 1996; Lindeberg and
Wessel-Berg, 1997; Riaz et al., 2006]. As a result, convective dissolution will eventually arrest the migration of
the buoyant CO2 current [Gasda et al., 2011; MacMinn et al., 2011; MacMinn et al., 2012; MacMinn and Juanes,
2013].
We now apply our model to demonstrate the implications of capillary pinning and blunting in the context
of geologic CO2 sequestration. Specifically, we compare the trapping efficiency of capillary pinning and
blunting with the combined effect of residual trapping and convective dissolution. MacMinn et al. [2011]
developed a sharp-interface model for the updip migration of a buoyant plume, under the influence of
residual trapping and convective dissolution:
~ @g 1 @ Ns ð12f 0 Þg2ð12f 0 Þg @g 52RN
~ d;
R
@s @f
@f
(13)
0
where g5h1 =H; f5x=l; f 5Mg=½ðM21Þg11, and s5t=T. The model is uniquely characterized by four
dimensionless parameters:
1. Ns 5ðl=HÞtan h, which measures the importance of upslope migration.
2. Nd 5l 2 qd =ðH2 /jMcos hÞ, which measures the importance of convective dissolution. The convective dissolution flux per unit area of fluid-fluid interface is qd 5avv Dqd gk=l2 , where vv measures the maximum
equivalent volume of free-phase CO2 that can dissolve in one unit volume of brine, Dqd is the density difference between brine and CO2-saturated brine, and a 0:01 is a constant. The value of Nd is not dependent
on permeability k, since j and qd are both proportional to k.
~ measures the importance of residual trapping. R512C
~
3. R
or 1 locally, depending on whether that portion of the interface is in imbibition or drainage, respectively. The residual trapping number C5Sgr =ð12Swc Þ
is a function of the residual gas saturation Sgr and the connate water saturation Swc.
4. M, the mobility ratio.
We apply the model with capillary pinning and blunting developed here, as well as the model with
residual trapping and convective dissolution developed by MacMinn et al. [2011] to four geologic
formations associated with actual CO2 sequestration projects. We set aquifer thickness H, permeability k,
residual gas saturation Sgr, and connate water saturation Swc to values reported in the literature (Table 3).
We consider a square-shaped instantaneous release of CO2 such that l 5 H. For simplicity, we fix the
aquifer slope h51o , pore-scale disorder f 5 1, and the fluid properties of CO2 and brine to values
representative of what could be encountered in deep saline aquifers (Table 4). We obtain
~ vary based on aquifer-specific properties.
M 13; Ns 0:017; Nd 7:531027 , while Bo21 and R
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Table 4. Fluid Properties Representative of What Could Be Encountered in Deep Saline Aquifers
qCO2 ðkg m23 Þ
700
qbrine ðkg m23 Þ
1000
lCO2 ðPa sÞ
6310
25
lbrine ðPa sÞ
8310
24
Dqd ðkg m23 Þ
vv (–)
Dq ðkg m23 Þ
M (–)
6
0.05
300
13
We use the stopping distance of the CO2 plume, xf, as a measure to compare the trapping efficiency
between the combined effect of capillary pinning and blunting, and the combined effect of residual trapping and convective dissolution (Figure 10). As expected, the effect of capillary pinning and blunting is significant for thin, low-permeability aquifers (i.e., high Bo21), such as the B-sandstone in the Tensleep
Formation in the Teapot Dome. Surprisingly, even at thick, high-permeability aquifers (i.e., low Bo21), capillary pinning and blunting could still be more effective in stopping the CO2 plume than residual trapping
and convective dissolution, as is the case for the Tuscaloosa Formation in the U.S. and the Naylor Field in
the CO2CRC Otway Project in Australia. In other cases, however, residual trapping and convective dissolution yield a shorter stopping distance. This is the case for the Mount Simon Sandstone in the U.S., which has
~ in addition to being thick and permeable (low Bo21).
a high residual trapping number C (low R),
Szulczewski et al. [2012] studied eleven of the largest aquifers in the conterminous U.S. as an effort to estimate the CO2 storage capacity in the country. We find Bo21 2 ½131024 ; 231022 in this group of thick and
permeable aquifers. Szulczewski et al. [2012] estimated C50:5 for all the aquifers they studied because
aquifer-specific data on the multiphase flow characteristics of CO2 and brine were largely unavailable. Bennion and Bachu [2008] conducted a series of relative permeability measurements using CO2 and brine in
sandstone samples from Alberta, Canada under reservoir conditions. We combine the values of Sgr and Swc
that they reported with the formations that we considered here and find C 2 ½0:33; 0:5. Hence, the majority
of large-scale CO2 sequestration projects will likely involve formations that lie in the upper left quadrant of
Figure 10. Our analysis shows that capillary pinning and blunting could be an important mechanism in limiting the migration distance of the CO2 gravity current.
We have assumed here that the aquifer is homogeneous, but all natural rocks are heterogeneous at some
scale. Further, the variety of complex pore geometries in natural rocks will lead to quantitatively different
results—for example, in terms of the scaling constant for the pinned interface height, and the pore-scale
disorder parameter. Quantifying
these parameters for different
rock types is beyond the scope of
this study, but these considerations are unlikely to have a strong
impact on our qualitative findings. It would be useful to include
residual trapping, solubility trapping, and capillary pinning and
blunting into a single model for
CO2 migration and trapping. We
have not done so here because
the coupling between these
mechanisms is not trivial. Residual
trapping occurs along the trailing
edge of the plume (i.e., the imbibition front), so we do not expect
it to have a strong interaction
Figure 10. Relative importance of capillary pinning and blunting versus residual trapping and convective dissolution in geological CO2 sequestration. We consider a squarewith capillary pinning at the
shaped instantaneous release of CO2 in a weakly sloped aquifer (h51o ) and use the
pinned interface or along the
dimensionless stopping distance xf =H as the metric of comparison. For capillary pinning
drainage front. Solubility trapand blunting, we use the model developed in this paper to obtain xf =H as a function of
ping, however, occurs along both
Bo21 (solid black line). The symbols show the dimensionless stopping distance of the
CO2 current under the influence of only residual trapping and convective dissolution, as
the imbibition front and the
predicted by the model developed by MacMinn et al. [2011], using parameters obtained
drainage front, and the way in
from aquifers associated with real CO2 sequestration projects. Our analysis shows that
which it interacts with capillary
capillary pinning and blunting could be just as effective as residual trapping and convective dissolution in limiting the ultimate migration distance of CO2.
pinning is unclear.
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7. Conclusions
We have shown via laboratory-scale experiments that capillary pinning stops the migration of immiscible
gravity currents at a finite distance. In addition, capillarity at the drainage front causes the buoyant, nonwetting current to thicken, a phenomenon that we refer to here as ‘‘capillary blunting.’’ Using experiments in
micromodels designed with pore-throat heterogeneity and anisotropy, we find that capillary pinning is
caused by capillary pressure hysteresis between drainage and imbibition while capillary blunting is caused
by the spatial distribution of capillary entry pressures on the drainage front. Both the length of the pinned
interface and the thickness of the nose scale with the relative importance between capillarity and gravity, as
described by the inverse Bond number, Bo21. Despite the strong influence of capillarity in immiscible gravity
currents, the nose position of the buoyant current with respect to time follows the same scaling as that of
miscible gravity currents prior to stopping, although capillarity does slow down the migration of the current.
We have developed a sharp-interface model that captures the effect of capillary pinning and blunting on
the evolution of immiscible gravity currents. We apply this model to aquifers associated with actual CO2
sequestration projects and find that capillary pinning and blunting could be an important mechanism in
limiting the maximum extent of CO2 migration, even in aquifers that are thick and permeable (i.e., low
Bo21). The reduction in maximum migration distance could reduce the risks of CO2 leakage through fractures or faults present in the aquifer caprock.
Appendix A: Numerical Implementation
We solve equation (12) for the interface height h, subject to no-flow boundary conditions at the ends of the
flow domain. We approximate the step-like initial configuration of h 5 0 for x < 0 and h 5 1 for x > 0 with
the smooth function hðxÞ512½tanhðbxÞ11=2, which approaches a unit step for large b. We solve the equation numerically using a centered finite volume method in space with forward Euler time integration
[Richtmyer and Morton, 1967].
The input parameters for the model are properties of the flow cell (porosity, permeability, and dimensions),
the properties of the fluids (densities and viscosities), the connate water saturation Swc, the strength of capillary pressure hysteresis in the system, as quantified by Dhc , and the characteristic nose height hn.
We discretize the spatial domain into N cells of size dx5ðxR 2xL Þ=N, denoted with index i51; 2; :::; N21; N.
xL and xR are the x coordinates of the left and right boundaries of the flow domain. Integrating equation
(12) over a cell, ½xi212 ; xi112 gives the integral form of the equation for each cell i:
ð i11 2
@h @ ð12f Þ
@h
@hc
h
dx50:
(A1)
2
1tan h2
@x
@t @x ð12Swc Þ @x
i212
We have a step-like initial condition such that the porous medium to the left of the initial interface is completely filled with the nonwetting fluid, and the connate water saturation in this region is assumed to be
dr
always zero (i.e., Swc 50; x < 0). Since we assume Pc to be the minimum drainage capillary pressure Pc;min
along the active drainage front where 12h hn , we get @hc =@x50 for ndr < i < nnose , where ndr is the cell
i
closest to the pinned interface that is experiencing drainage (i.e., dh
dt < 0) and nnose is the cell where
12h < hn . Similarly, since we assume Pc to be the characteristic imbibition capillary pressure on the imbibition front, @hc =@x50 for i < nimb , where nimb is the cell closest to the pinned interface that is experiencing
i
imbibition (i.e., dh
dt > 0). The interface between ndr and nimb is pinned. On the drainage front and the imbibition front, equation (A1) reduces to
ð i11 2
@h @ ð12f Þ
@h
h
2
1tan h
dx50:
@t @x ð12Swc Þ @x
i212
(A2)
Approximating the gravity current thickness as piecewise-constant (constant over each cell i), we have
i112
@hi
dx1fFðhÞg 50;
@t
i21
(A3)
2
where F(h) is the flux function evaluated at the intercell edges ði1 12Þ and is given by
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FðhÞ52
ð12f Þ
@h
h
1tan h :
ð12Swc Þ @x
(A4)
We use a two-point flux approximation of equation (A4),
Fi112 ð12f Þ
hi11 2hi
2
h
1tan h :
dx
ð12Swc Þ i11
(A5)
2
The term in the curly bracket in equation (A5) plays the role of an effective diffusion coefficient at the interface, Di112 . To calculate this effective diffusion coefficient at the interface, we take the arithmetic mean of
the diffusion coefficients at the adjacent grid blocks
Di112 Di 1Di11
:
2
(A6)
Across the pinned interface, we have
hdr 2h1
1tan h ;
dx=2
2
h
2himb
F 2 Dimb
1tan h ;
dx=2
F 1 Ddr
(A7a)
(A7b)
where F1 and F– are the fluxes on the drainage side and the imbibition side of the pinned interface, respectively (Figure 7). For simplicity, we use hdr and himb to approximate the nonlinear diffusion coefficients Ddr
and Dimb . By definition, we have h1 2h2 5Dhc . Since h does not change in time along the pinned interface,
F1 must equal to F– by volume conservation. We solve for h– by enforcing F 1 5F 2
Dimb himb 2 dx2 tan h 1Ddr hdr 2Dhc 1 dx2 tan h
2
h 5
;
(A8)
Ddr 1Dimb
and obtain the flux across the pinned interface by substituting h– into equation (A7).
At the nose of the current where 12h < hn , we have Fnose 50 as the difference in drainage capillary pressure
here is balanced by hydrostatic pressure.
To discretize the equation in time, we employ the forward Euler method, so that
in
dt h
hn11
5hni 1
Fi112 2Fi212 :
i
dx
(A9)
We update ndr and nimb after each time step. For drainage, a portion of the active drainage front becomes
pinned when it is no longer draining (i.e., dh
dt 0). For imbibition, the imbibition front starts to extend outward when the entire initial vertical interface is depinned.
Acknowledgments
We thank Michelle Dutt for assistance
with the experiments and Michael
Szulczewski and Jerome Neufeld for
comments. This work was partly
funded by the US Department of
Energy (grants DE-SC0003907 and
DE-FE0002041) and the MIT/Masdar
Institute Program. Additional funding
was provided by the MIT Energy
Fellows Program (to B.Z.), a Yale
Climate and Energy Institute
Postdoctoral Fellowship (to C.W.M.), a
Martin Fellowship for Sustainability (to
M.L.S.), the Royal Society Wolfson
Research Merit Award (to H.E.H.), and
the ARCO Chair in Energy Studies (to
R.J.).
ZHAO ET AL.
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